An important thing is that the representation tensor categories (with conjugates) of \(U_q\mathfrak {gl}(n)\)  and \(\mathrm {U}_q(n)\)  are naturally identified with each other; see [1]}. Hence the algebras \(W^*(\mathrm {U}_q(n)) \subset \mathcal {U}(\mathrm {U}_q(n))\)  can directly be constructed from \(U_q\mathfrak {gl}(n)\)  (or more precisely, its representation category).
Another important fact is that the special positive element \(\rho _n \in \mathcal {U}(\mathrm {U}_q(n))\)  (see (REF )) is given by
\(\rho _n = K_1^{-n+1} K_2^{-n+3} \cdots K_n^{n-1} \in U_q\mathfrak {gl}(n).\) 
